The ring of integers of
\(K_m\)  is principal for \(m\in \lbrace 3, 4, 5\rbrace \)  (see [1]}). So \(h(K_m^+)=1\) . By [2]} and Lemma REF , the unit group \(E_{K_m}\)  of \(K_m\)  is generated by \(\zeta _{ 2^{m}}\)  and \(\xi _{k, m}=\zeta _{2^m}^{(1-k)/2}\frac{1-\zeta _{2^m}^k}{ 1-\zeta _{2^m}}, \)  where \(k\)  is an odd integer such that \(1< k< 2^{m-1}\) . So by the ambiguous class number formula (see [3]}) and Proposition REF ,
we have \(rank_2(Cl(L_{m, d}))=2r-1-e_{m, d}\) . Let \(p\)  be a prime divisor of \(d\)  and \( \mathfrak {p}_{K_m}\)  a prime ideal of \(K_m\)  lying over \(p\) .
If all the prime divisors of \(d\)  are in the same coset \(\pmod {8}\) , then by Lemma REF  it is easy to see that
\(E_{K_m}/(E_{K_m}\cap N(L_{m, d}))=\lbrace \overline{1}, \overline{\zeta _{2^m}}\rbrace \) . Hence \(e_{m, d}=1\) .
Suppose now that the prime divisors of \(d\)  are not in the same coset \(\pmod {8}\) . By Lemma REF , we have \(\xi _{k, m}\)  is a norm in \(L_{m, d}/K_m\) , for all \(k\equiv \pm 1\pmod {8}\) .
Let \(k\ne k^{\prime }\)  be two odd positive integers such that \(1< k, k^{\prime }< 2^{m-1}\)  and \(k, k^{\prime }\lnot \equiv \pm 1 \pmod {8}\) . Again by Lemma REF , we have:
\(\left( \frac{\xi _{k, m}\xi _{k^{\prime }, m}, d}{\mathfrak {p}_{K_m}}\right)=1 \text{ for all $\mathfrak {p}_{K_m}$ of $K_m$, } \) 
and:
\(\left( \frac{\zeta _{ 2^{m}}\xi _{k, m}, d}{\mathfrak {p}_{\mathfrak {p}_{K_3}}}\right)=-1\; \text{ if } \mathfrak {p}_{K_m} \text{ is lying over $p\equiv 5\pmod {8}$}.\) 
So \(\overline{\xi _{k, m}}=\overline{\xi _{k^{\prime }, m}}\)  and \(\overline{\xi _{k, m}}\ne \overline{\zeta _{ 2^{m}}}\)  in \(E_{K_m}/(E_{K_m}\cap N(L_{m, d}))\) . Thus \(E_{K_m}/(E_{K_m}\cap N(L_{m, d}))= \lbrace \overline{1}, \overline{\zeta _{ 2^{m}}}, \overline{\xi _{k, m}}, \overline{\zeta _{ 2^{m}}\xi _{k, m}}\rbrace \) . Hence \(e_{m, d}=2\) . So we have the theorem for \(m\in \lbrace 3, 4, 5\rbrace \) .
Let \(\pi _1= 2\) , \(\pi _2=2+\sqrt{2}\) , ..., \(\pi _m=2+\sqrt{\pi _{m}}\) . Set \({k}=\mathbb {Q}(\sqrt{d}, \sqrt{-1})\)  and
\({k}_1={k}(\sqrt{\pi _1})=L_{3, d}\) , \({k}_2={k}(\sqrt{\pi _2})=L_{2, d}\) , ..., \({k}_m={k}(\sqrt{\pi _m})=L_{m, d}\) . Thus, the cyclotomic \(\mathbb {Z}_2\) -extension \({k}_{\infty }\)  of \({k}\)  is given by \(\cup _{m=0}^{\infty }{k}_m\) . As we have proved Theorem REF  for
the three layers \({k}_1\) , \({k}_2\)  and \({k}_3\) , then Theorem REF  achieves the proof.
